Three-Phase Three-Level Flying Capacitor PV Generation ...

Electronics 2019, 8, 118; doi:10.3390/electronics8020118 www.mdpi.com/journal/electronics

Article

Three-Phase Three-Level Flying Capacitor PV Generation System with an Embedded Ripple Correlation Control MPPT Algorithm

Manel Hammami 1, Mattia Ricco 1, Alex Ruderman 2 and Gabriele Grandi 1,*

1 Department of Electrical, Electronic, and Information Engineering, University of Bologna,

Viale Risorgimento 2, 40136 Bologna, Italy; [email protected] (M.H.); [email protected] (M.R.) 2 Department of Electrical and Computer Engineering, Nazarbayev University, 53 Kabanbay Batyr Ave,

Astana 010000, Kazakhstan; [email protected]

* Correspondence: [email protected], Tel.: +39-051-20-93571

Received: 19 December 2018; Accepted: 21 January 2019; Published: 22 January 2019

Abstract: This paper presents the implementation of a maximum power point tracking (MPPT)

algorithm in a multilevel three-phase photovoltaic (PV) system using the ripple correlation control

(RCC) method. Basically, RCC adopts the inherent oscillations of PV current and voltage as

perturbation, and it has been predominantly used for single-phase configurations, where the

oscillations correspond to the 2nd order harmonics. The implementation of an RCC-MPPT

algorithm in a three-phase system has not been presented yet in the literature. In this paper, the

considered three-phase three-level converter is a three-level flying capacitor (FC) inverter. The

proffered RCC method uses the 3rd harmonic components of PV current and voltage for the

estimation of the voltage derivative of the power dPpv/dVpv (or current, dIpv/dVpv), compelling the PV

array to operate at or very close to the maximum power point. The analysis and calculation of the

low-frequency PV current and voltage ripple harmonic components in the three-phase flying

capacitor inverter is presented first, with reference to centered carrier-based three-level PWM. The

whole grid-connected PV generation scheme has been implemented by MATLAB/Simulink, and

detailed numerical simulations verified the effectiveness of the control method in both steady-state

and dynamic conditions, emulating different sun irradiance transients.

Keywords: Photovoltaic; three-level flying capacitor inverter; three-phase inverter; maximum

power point (MPP); ripple correlation control (RCC); low-frequency harmonics

1. Introduction

To fulfill the constantly increasing worldwide energy demand, renewable energy sources such

as photovoltaic (PV), wind, geothermal, and biomass, are being explored. Currently, PV energy is

becoming one of the most widely used renewable energy sources, due to its indubitable known

advantages and the decreasing installation costs.

In order to improve the PV conversion efficiency, maximum power point tracking (MPPT)

algorithms are widely adopted in both grid-connected and stand-alone PV systems. Among the

different methods known in the literature, the most popular and effective are perturb and observe

(P&O) [1] and incremental conductance [2]. Typical problems for these methods are the

identification of a suitable perturbation step size, and the limited maximum power point (MPP)

tracking dynamic capability during sudden variations in solar irradiance. In this class of P&O

algorithms, either fixed or variable step size are adopted to improve the settling time in transient

conditions and the MPP resolution in the steady-state operating points [3].

Similar to P&O techniques, the extremum seeking (ES) algorithm employs “ad hoc” perturba-

tions [4]. A multi-variable extremum seeking algorithm based on a single control loop for cascaded

Electronics 2019, 8, 118 2 of 15

dc/dc photovoltaic micro-converters has been proposed in [5]. A Newton-based ES algorithm has

been adopted to improve the dynamic performances of gradient-based ES in [6]. All aforementioned

algorithms implemented the ES control by injecting an external perturbation signal into the duty

cycle. On the contrary, ripple correlation control (RCC) algorithms exploit the inherent PV voltage

and current oscillations to track the MPP. In particular, for single-phase PV systems the 2nd

harmonics are exploited [4,7–13]. The RCC algorithm has good dynamic performance comparing

with the P&O algorithms and it does not require additional perturbations in tracking the MPP. A

comprehensive analysis and comparison between RCC and ES methods has been carried out in

reference [9] in case MPPT algorithms for PV systems.

The basic RCC-MPPT method has been investigated in references [9,12,14] where two low-pass

filters and two high-pass filters are used for ripple extraction and implementation. In reference [10],

a modified RCC method has been proposed by using the moving average concept instead of

high/low-pass filtering to improve the dynamic response, without the need to tune the time constant

of the low/high pass filters. Moreover, the scheme has been simplified by using only the sign of the

product of power and voltage ripple to drive the PV operating point toward the MPP. In reference

[13], a hybrid RCC-MPPT algorithm has been proposed to improve stability during sudden solar

irradiance transients. In case of three-phase two-level (2L) inverters, the RCC algorithm cannot be

applied due to the inexistence of inherent (natural) low-order harmonic oscillations on the PV side of

the inverter. In this case, the ES algorithm can be successfully applied by means of additional

perturbations, as reported in reference [15].

In recent years, multilevel inverters have become more attractive for single- and three-phase

systems thanks to their advantages over the conventional inverters [7,15–19]. They offer improved

output waveforms, lower total harmonic distortion (THD) and smaller grid filter size [20–22]. The

most common multilevel converter configurations, presented in literature, are the cascaded

H-bridge (CHB), neutral-point-clamped (NPC) and flying capacitor (FC). These types of converters

are also adopted in PV applications due to the aforementioned advantages. Multilevel flying

capacitor inverters have some specific distinct advantages over the other two topologies such as: no

need for many isolated dc sources when compared to CHB inverters, no need for clamping diodes in

contrast to NPC inverters, but still preserving the ability to self-balance the capacitor voltages.

With reference to the FC inverter, several analyses have been presented in the literature. A

mathematical model for the dynamic behavior of a simple FC inverter using a sampled-data

modeling approach has been derived in reference [23], with useful information on the power circuit

characteristics and its natural balancing property. In reference [24], a modified pulse with

modulation (PWM) strategy is introduced to improve the balancing rate of capacitor voltages,

mainly for small output voltages, by optimizing the use of redundancy of switching states. A new

PWM scheme has been proposed and analyzed in reference [25], which results in better balancing

properties than the normal phase-shifted (PS) PWM. In particular, a five-level configuration has

been considered and a modified PS-PWM scheme has been studied, solving the slow-balancing

problems of the normal PS-PWM method for odd-level of FC converters.

In general, adopting a multilevel inverter introduces different low-order voltage and current

harmonics on the dc-link side compared to the case of a single-phase H-bridge inverter where only

the 2nd order harmonics are present. The application of the RCC algorithm in case of a single-phase

inverter with level doubling network (LDN) has been introduced and examined in reference [10].

Due to the multiple harmonics, the basic implementation of the RCC-MPPT scheme becomes

deficient, leading to a misestimating of the voltage derivative of the power (dPpv/dVpv). For this

reason, a modified RCC scheme extracting the amplitude of a definite harmonic form PV voltage

and current (dc-link) waveforms has been proposed. In order to maximize the resolution, the

proposed solution makes reference to the highest amplitude harmonic, leading to a more effective

estimation of dPpv/dVpv.

Though numerous RCC-MPPT algorithms for single-phase grid-connected PV systems have

been developed, no analysis of RCC-MPPT in case of three-phase multilevel inverters has been

reported. This paper presents an RCC-MPPT algorithm for a grid-connected three-phase three-level


FC inverter (Figure 1) with reference to centered level-shifted (LS) carrier-based PWM modulation.

Moreover, a complete analysis of PV voltage and PV current harmonic components is accomplished,

being the foundation of the ripple correlation control. In this case, dc-link voltage and current

harmonics are introduced by the instantaneous power oscillations between the three flying

capacitors and the dc bus, consisting mainly of 3rd harmonic components. These voltage and current

harmonics are exploited as embedded perturbations to determine the MPP of the PV array. The

considered grid-connected three-phase PV generation system is presented in the block diagram of

Figure 1.

fL

pvi

pvV

Figure 1. Three-phase grid-connected PV generation system based on three-level FC inverter.

2. Modulation Principle for the Three-Level FC Inverter

With reference to linear and balanced sinusoidal modulation, the modulating signals (uA, uB,

and uC) correspond to the inverter output voltages (vA, vB, and vC) normalized by Vpv and averaged

over the switching period (Tsw =1/fsw):

mm

mm

mm

CuCmu

CuCmu

CuCmu

*C3

4C

*B3

2B

*AA

)(sin

)(sin

)(sin

, (1)

where ϑ = ωt is the phase angle, ω is the fundamental (angular) frequency, m is the modulation

index, *iu are the reference (normalized) output voltages (i = A, B or C), and Cm is the

common-mode signal to maximize the linear modulation range.

The voltages across the flying capacitors CA, CB and CC are spontaneously regulated to the half

of the DC-link voltage if a proper modulation technique with self-balancing capability is adopted

[19]. Introducing for the upper switch of each phase i (i = A, B, C) the switching function )1(

iS

(averaging is denoted in the following by overline) results in:

1)1( iii uuS . (2)

With reference to phase A, considering Equations (1) and (2), it becomes:

1)(sin)(sin)1(

A mm CmCmS . (3)

The switching functions for the other phases B and C are readily obtained by exploiting the

modulation symmetry among the three phases, according to Equation (1).

In order to improve the so-called sinusoidal PWM (SPWM), in which there is no common-mode

signal injection, Cm = 0, one of the most popular ways to maximize the modulation index is the so

called “centered” PWM (CPWM), consisting of the injection of a common-mode signal able to center

the modulating signals:


),,min(),,max(2

1 *C

*B

*A

*C

*B

*A uuuuuuCm . (4)

In case of sinusoidal reference voltages, Equation (1), Cm can be rewritten as:

6

11

2

3,

6

5

2,

3

2sin

2

3

6

7,

26,

3

2sin

6

7

6

5,

66,sin

2

1mCm . (5)

Figure 2 shows an example of the considered carrier-based CPWM strategy for three different

modulation indices.

Si(2), Si

(3)

Si(1), Si

(4)

ui

Figure 2. Carrier-based CPWM modulation logic for each leg (i) of three-level FC inverter in case of m

= 0.577/2 (red trace), m = 0.43 (green trace) and m = 0.577 (blue trace). Underline denotes

complementary.

3. Evaluation of PV Current and Voltage Harmonics

In the studied single-stage PV generation system (Figure 1), the PV array is directly connected

to the dc-link bus of the three-phase FC inverter. In this way, PV voltage harmonics correspond to

the inverter dc-link voltage harmonics, whereas PV current harmonics can be calculated on the basis

of the inverter dc-link current harmonics by the following procedure.

3.1. Inverter Input Current Harmonics

The instantaneous inverter input current i(t) consists of the averaged component over the

switching period i , and the instantaneous switching ripple component i . Similarly, the current

averaged over the switching period consists of its averaged component over the fundamental

period Idc (dc component) and the alternating low-frequency harmonic component i~

. These

compositions are summarized as:

iiIiiti dc ~

)( . (6)

Neglecting the switching ripple, the output currents can be considered a sinusoidal balanced

system with the amplitude Iac and the general output phase angle comparing to grid voltages:

)(sin

)(sin

(sin

34

c

32

b

a

ac

ac

ac

Ii

Ii

Ii

. (7)

For each inverter leg, the averaged input current (over the switching period) can be determined

by multiplying the switching function of upper switch by the corresponding output current. With

reference to leg A, it leads to:

aAA)1( iSi . (8)


By introducing Equations (3) and (7) in Equation (8), the averaged current of the leg A becomes:

]1)(sin)(sin[(sinA mmac CmCmIi . (9)

The resulting averaged total input current i is the sum of the three leg currents:

cba iSiSiSi )1()1()1(

CBA . (10)

In the following, the analysis is restricted to the phase angle span 2/3 since the inverter input

current has a periodicity of T/3. In particular, with reference to the phase angle range 2/3 ≤ ≤ 4/3,

the inverter input current can be expressed in the following two sub-ranges /3:

3

4 ,

3

2 ,)1(

cCbaA

cC

)1()1(

)1(

iSiiS

iS

i . (11)

Introducing Equations (1), (2), and (7) in Equation (11) leads to:

3

4

6

7,cos5

3cos

2

32cos3

6

7 ,cos

2

33

2cos

2

1

6

72cos3

6

5,

6cos

62cos3

6

5

3

2 ,

6cos

2

32cos3

2acmI

i . (12)

By examining Equation (12), the average and the low-frequency input current components can

be determined in case of centered PWM as:

cos2

3acdc ImI , (13)

3

4

6

7,cos2

3cos

2

32cos3

6

7 ,cos

2

3

2cos

2

1

6

72cos3

6

5,)cos(3

6cos

62cos3

6

5

3

2 ,)cos(3

6cos

2

32cos3

2

~ acmIi . (14)

Although the above developments are carried out for centered PWM, the input inverter current

analysis can be similarly extended to other PWM techniques. As an example, the simpler case of

sinusoidal PWM has been presented in reference [21].

In order to prove the validity of Equations (12)–(14), some simulation tests concerning the input

inverter currents, with specific reference to unity power factor ( = 0°) and unity sinusoidal output

currents (Iac = 1A), are presented in Figure 3.

In particular, top traces in Figure 3 show the simulation results comparing the instantaneous

input current of leg A iA (blue trace) with its averaged value over the switching period Ai (red

trace), and the corresponding low-frequency current component calculated by replacing Equation

(5) in Equation (9) (green trace). Two cases of modulation index m = 0.577/2 (Figure 3, left) and m =

0.577 (Figure 3, right) are considered.


The bottom traces in Figure 3 show the total instantaneous inverter input current i(t) (blue

trace) and its averaged value over the switching period i (red trace). The averaged current

component (green trace) is determined analytically in case of centered PWM by Equation (12).

Figure 3 shows a perfect matching between theoretical and simulated results. The slight delay (Tsw/2)

between numerically averaged values and the corresponding theoretical averaged values is due to

the numerical averaging itself.

Figure 3. One leg input current (top) and total inverter input current (bottom): instantaneous value

(blue trace), its numerically averaged value over the switching period (red), and theoretical value

(green) in case of for m = 0.577/2 and m = 0.577 (from right to left), Iac = 1 A and

3.2. PV Voltage Harmonics

Similarly to the input dc-link current, dc-link voltage components can also be pointed out. The

instantaneous dc-link voltage vpv, corresponding to the PV voltage, consists of its averaged value

over the switching period pvv and the instantaneous switching ripple vpv. Also, the averaged

component over the fundamental period Vpv (dc component) and the low-frequency harmonic

component pvv~ can be introduced, leading to:

pv pv pv pv pv pv pv pvv v v V v v V v . (15)

The switching ripple component vpv is practically insignificant for switching frequencies

starting from few kHz, being completely smoothed by the dc-link capacitor. For this reason, it is

assumed here to be vpv ≈ 0. The alternating low-frequency voltage component can be evaluated

based on the low-frequency input current component i~

taking into account the whole dc-link

impedance, given by the parallel of the capacitor reactance and the equivalent resistance of the PV

array at maximum power point (Rpv ≈Vmpp/Impp).

Taking into consideration the expected parameters for PV arrays and dc-link capacitors, the

assumption Rpv >> 1/(kC) is usually true, with k being the harmonic order. In this case, the whole

alternating input current component is circulating through the dc-link capacitor, and the alternating

dc-link (PV) voltage component can be calculated by integrating Equation (14). The resulting four

contributions can be adopted in order to calculate the amplitude (peak value) of the alternating

low-frequency voltage component:


24,

13,

2,

,1

V~

V~

V~

V~

cos8

32sin

38V~

cos8

32sin

38V~

pv

pv

acpv

acpv

C

Im

C

Im

. (16)

As most of the grid-connected PV systems operate under unity power factor conditions, the

case = 0° is considered in the following developments. In this case, the peak of the alternating

low-frequency voltage ripple is calculated taking into account either the first or the second

contribution, leading to:

2 3V 0.04

8ac ac

pv

I Im m

C C

. (17)

Equation (17) presents the peak value of the alternating voltage ripple in case of CPWM with

reference to = 0. However, the peak value of the same quantity has been determined in case of

SPWM in reference [21] as:

V 0.167 acpv

Im

C . (18)

Comparing Equations (17) and (18), it should be noted that practically the peak value of the

alternating low-frequency voltage ripple in case of CPWM is almost four times less than the voltage

ripple in the case of SPWM.

As an example, the instantaneous PV voltage is shown in Figure 4 (green trace) together with its

averaged value over the switching period (pink trace) in case of Iac/C = 1 for two cases of

modulation indices m = 0.577/2 and m = 0.577. A small delay between averaged and instantaneous

values (Tsw/2) can be noticed, due to the averaging itself. The peak value of the alternating PV

voltage component corresponds to the result obtained by Equation (17). In particular, pvv~ is equal

to 0.011 V and 0.023 V for m = 0.577/2 and m = 0.577, respectively.

As expected, there is a very good matching between simulation and analytical results.

11 mV

0.22 Time (s)

23 mV

0.22 Time (s)

Figure 4. Alternating PV voltage ripple (green trace) and its averaged value over the switching

period (pink trace), for m = 0.577/2 (left) and m = 0.577 (right), Iac/wC = 1.

3.3. PV Current Harmonics

With regard to the PV voltage vpv (Equation 15) and neglecting the switching frequency

component, the instantaneous PV current ipv can be written as:

pvpvpv iIi~

, (19)


being Ipv the average over the fundamental period (Ipv = Idc, due to the dc-link capacitor) and pvi~

the

alternating low-frequency harmonic component, calculated as:

pvpvpv Rvi /~~ . (20)

The amplitude of the alternating low-frequency PV current ripple is calculated based on

Equations (17) and (20) as:

2 30.04

8ac ac

pv

pv pv

I II m m

R C R C

. (21)

As an example, Figure 5 shows the instantaneous PV current (red trace) together with its

averaged value over the switching period (blue trace) in case of Iac/C = 1 and for two cases of

modulation index, considering a PV series equivalent resistance Rpv = 7.13 , representing the scale

factor with Figure 4.

1.5 mA

Time (s)

3.2 mA

Time (s)

Figure 5. Alternating PV current ripple (red trace) and its averaged value over the switching period

(blue trace), for m = 0.577/2 (left) and m = 0.577 (right), Iac/wC = 1 and Rpv = 7.13 .

4. Proposed RCC-MPPT Algorithm

Figure 6 presents the block diagram of the adopted control scheme, including MPPT and

dc-link (PV) voltage regulation. In particular, the proposed RCC algorithm estimates the voltage

derivative of the power, dPpv/dVpv, used to drive the working point to the maximum power point.

Integrating dPpv/dVpv, the reference dc-link voltage vpv* is simply obtained and a proportional-integral

(PI) voltage regulator can be used to determine the reference grid current amplitude Iac*. In order to

inject a sinusoidal current into the grid with unity power factor, a traditional dq current controller

has been used.

PW

M

*acI

*ABCv

abcv abci

controllervoltage

PV

pvipvv

pvv

*pvv

pv

pv

dV

dPpvv

Figure 6. Control scheme of the PV generation system.

An example of PV voltage and current ripple in case of a three-phase FC inverter configuration

is given in the following. Figure 7 shows the results considering the PV conversion scheme for a


given operating point. As expected, the inverter voltage has nine levels for a higher modulation

index. Also, it is noticeably clear that PV voltage and current contain a huge 3rd harmonic component

(150 Hz) and are in phase opposition.

(a)

(b)

(c)

vA

ia

ipv

vpv

Time (s)

Figure 7. Simulation example of: (a) grid current and inverter output voltage, (b) PV voltage, and (c)

PV current, in case of three-phase flying capacitor inverter.

In case of three-phase PV systems, with the inverter being directly connected to the PV array,

the FC inherent 3rd (and odd multiple) order harmonic appears in PV current and PV voltage (as

shown in Figure 7). As is known, ripple correlation control algorithm is able to exploit the amplitude

of these oscillations to provide information about the operating point of the PV array. Considering

the working point Q, the voltage derivative of the PV power dPpv/dVpv can be s written as:

pv

Qpv

pvpv

Qpv

pvV

dV

dII

dV

dP , (22)

where dIpv/dVpv is defining the voltage derivative of the current as:

pv

Qpv

pvpv v

dV

dIi ~~

. (23)

A conventional RCC-MPPT algorithm is implemented by multiplying the voltage oscillation by

the current oscillation, and integrating over the harmonic period [10]. As an alternative

implementation, the estimation of dIpv/dVpv (and then dPpv/dVpv) can be carried out by considering a

specific harmonic order, which is the 3rd in the case of the FC inverter. Equation (23) can be then

rewritten as:

3,3,~~

pv

Qpv

pvpv v

dV

dIi . (24)

By following the approach of the conventional RCC-MPPT implemented in [10], the estimation

of the voltage derivative of the current dIpv/dVpv can be calculated, considering the third harmonic

component as:

Electronics 2019, 8, 118 10 of 15

3,

3,

23,

3,3,

23,

3,3,3

~

~~

pv

pv

pv

pvpv

t

Tt

pv

t

Tt

pvpv

Qpv

pv

V

I

V

VI

dtv

dtvi

dV

dI

. (25)

It should be noted that dIpv/dVpv derived by Equation (25) is calculated by integrating the

product of the PV voltage and current over the fundamental period T, as defined by Fourier series

(FS) to calculate the 3rd harmonic. The block diagram of Figure 8 shows a possible implementation of

Equation (25), with the estimation of dPpv/dVpv considering only the 3rd harmonic component.

LPF

3,pvI

3,pvV

harmonic

FS

rd3

harmonic

FS

rd3

pvi

pvv

pvi~

pv

pv

dV

dP

pv

pv

dV

dI

pvV

pvI

pvV

pvI

pvv~

Hz

MAvg

150

Hz

MAvg

150

Figure 8. Block diagram of proposed RCC, estimating dPpv/dVpv by the 3rd harmonic component.

5. Simulation Results

In order to validate the proposed analysis, the three-phase grid-connected PV generation

system shown in Figure 1 was simulated in Matlab-Simulink. The circuit parameters are

summarized in Table 1. A string of series/parallel connected PV modules has been adopted as PV

source, according to the data given in Table 2 in standard test conditions (STC). In particular, the

considered PV modules are SP-305 type (96 cells, monocrystalline).

Table 1. Simulation circuit parameters.

Label Description Parameters

Vac Grid voltage (rms) 230 V

Lf, Rf Ac-link inductor 1 mH, 3 m

C DC-link capacitor 2 mF

CA, CB, CC Flying capacitors 5 mF

f, fsw Fundamental and switching frequencies 50 Hz, 3kHz

Table 2. PV module specifications (STC) and modules arrangement.

Label Description Parameters

VOC Open circuit voltage 64.2 V

ISC Short circuit current 5.96 A

Vmpp Maximum photovoltaic voltage 54.7 V

Impp Maximum photovoltaic current 5.58 A

NS Number of series-connected modules per string 16

NP Number of parallel strings 22

The first simulation test is carried out in order to verify the input/output steady-state

waveforms of the PV generation system with considered three-phase three-level FC inverter.

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In particular, Figure 9 shows grid voltage and current (top traces), PV voltage (medium trace)

and PV current (bottom trace). It can be noticed that the grid current is almost sinusoidal, with unity

power factor. As expected, the PV voltage and PV current have 3rd harmonic component oscillations

(150 Hz), in phase opposition. The proposed RCC-MPPT algorithm can well track the maximum

power point at sun irradiance E = 1000 W/m2 (in this case Vmpp = 875 V and Impp = 123 A).

Figure 9. Input/output steady-state waveforms of the FC in case of sun irradiance E = 1000 W/m2:

grid voltage and current (top traces), PV voltage (middle trace) and PV current (bottom trace).

The following simulation tests are carried out to verify the dynamic performance of the

proposed RCC-MPPT algorithm in case of fast solar irradiance transients. In particular, two

scenarios have been considered:

1. Sun irradiance (ramp) increase,

2. Sun irradiance (ramp) decrease.

The cell temperature has been considered constant (25°C) during simulations for both

scenarios.

The simulation results depicted in Figures 10 and 11 show the behavior of the considered

system during the first scenario. A linear ramp increase of solar irradiance from 500 W/m2 to 1000

W/m2 is applied in a transient period of 200 ms, between 1.3 s and 1.5 s. The results clearly indicate

that as soon as the transient occurs, the estimation of dPpv/dVpv is oscillating. However, it is almost

acceptable during the steady-state (Figure 10b), which means that the PV current and the PV voltage

perfectly follow the MPP under this kind of smooth irradiance change (Figure 10c and d). Figure 10e

presents the waveform of the injected grid current. The injected grid current is perfectly sinusoidal

and well controlled thanks to the dynamic performance of the dq current controller.

Figure 11 shows the Ppv(Vpv) and Ipv(Vpv) diagrams corresponding to the transient depicted in

Figure 10. The operating point moves from the first MPP, before the solar irradiance transient, to the

new MPP, in the steady-state condition after the transient.

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Figure 10. Irradiance ramp increase: (a) irradiance, (b) power derivative, (c) photovoltaic current. (d)

photovoltaic voltage, and (e) grid current, in case of proposed RCC-MPPT algorithm.

Figure 11. Effects of a 50% irradiance transient of PV power vs. PV voltage in case of irradiance ramp

decrease.

Simulation results for ramp decrease in solar irradiance are presented in Figure 12 and 13. In

this case, the initial value of solar irradiance is 1000 W/m2, decreasing linearly to 500 W/m2 in a

period of 200 ms. As can be seen, the estimation of dPpv/dVpv is correct during the steady state (Figure

12b). At the end of the transient, the estimation of dPpv/dVpv becomes correct again, and the operating

point is correctly driven toward the MPP. The current injected into the grid is perfectly sinusoidal

and follows the sun irradiance profile (Figure 12e). The path of the operating points is displayed on

the Ppv(Vpv) and Ipv(Vpv) diagrams and presented in Figure 13.

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Figure 12. Irradiance ramp decrease: (a) irradiance, (b) power derivative, (c) photovoltaic current, (d)

photovoltaic voltage, and (e) grid current, in case of proposed RCC-MPPT algorithm.

Figure 13. Effects of a 50% irradiance transient of PV power vs. PV voltage in case of irradiance ramp

decrease.

6. Conclusion

In this paper, an original RCC-MPPT algorithm suitable for three-phase three-level flying

capacitor photovoltaic systems has been proposed and analyzed in detail. The three-phase FC

inverter introduces voltage and current harmonics on the input (i.e., PV) side. In particular, a

significant 3rd order harmonic is clearly noticeable, whereas higher harmonic components have

much lower amplitudes. Both PV voltage and current harmonic amplitudes are analytically

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calculated in the whole modulation range in case of centered PWM, offering the possibility of a

precise and effective design of the DC-link capacitor in order to obtain the desired ripple amplitude.

Because of the presence of different harmonics, a ripple correlation control scheme extracting

the amplitude of a specific harmonic from PV voltage and current waveforms has been proposed in

order to track the maximum power point of the PV arrays. Specifically, the estimation of dIpv/dVpv

(and then dPpv/dVpv) is carried out by considering the 3rd order harmonic of PV voltage and current

oscillations.

Numerical tests have been performed to prove the effectiveness of the whole PV generation

scheme, including three-phase three-level FC inverter and the proposed RCC-MPPT algorithm.

Tests have been carried out considering both steady-state conditions and fast solar irradiance

transients in order to verify the dynamic performance of the proposed RCC algorithm, resulting in

an effective and original MPPT method in the case of three-phase PV systems.

Author Contributions: M.H. and G.G. developed the theoretical analysis; M.H. performed the simulation

results, also providing for the manuscript arrangement in cooperation with M.R.; A.R. and G.G. generally

supervised and finalized the work.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

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